108,445 research outputs found
Systematic Analysis of Majorization in Quantum Algorithms
Motivated by the need to uncover some underlying mathematical structure of
optimal quantum computation, we carry out a systematic analysis of a wide
variety of quantum algorithms from the majorization theory point of view. We
conclude that step-by-step majorization is found in the known instances of fast
and efficient algorithms, namely in the quantum Fourier transform, in Grover's
algorithm, in the hidden affine function problem, in searching by quantum
adiabatic evolution and in deterministic quantum walks in continuous time
solving a classically hard problem. On the other hand, the optimal quantum
algorithm for parity determination, which does not provide any computational
speed-up, does not show step-by-step majorization. Lack of both speed-up and
step-by-step majorization is also a feature of the adiabatic quantum algorithm
solving the 2-SAT ``ring of agrees'' problem. Furthermore, the quantum
algorithm for the hidden affine function problem does not make use of any
entanglement while it does obey majorization. All the above results give
support to a step-by-step Majorization Principle necessary for optimal quantum
computation.Comment: 15 pages, 14 figures, final versio
ROM-based quantum computation: Experimental explorations using Nuclear Magnetic Resonance, and future prospects
ROM-based quantum computation (QC) is an alternative to oracle-based QC. It
has the advantages of being less ``magical'', and being more suited to
implementing space-efficient computation (i.e. computation using the minimum
number of writable qubits). Here we consider a number of small (one and
two-qubit) quantum algorithms illustrating different aspects of ROM-based QC.
They are: (a) a one-qubit algorithm to solve the Deutsch problem; (b) a
one-qubit binary multiplication algorithm; (c) a two-qubit controlled binary
multiplication algorithm; and (d) a two-qubit ROM-based version of the
Deutsch-Jozsa algorithm. For each algorithm we present experimental
verification using NMR ensemble QC. The average fidelities for the
implementation were in the ranges 0.9 - 0.97 for the one-qubit algorithms, and
0.84 - 0.94 for the two-qubit algorithms. We conclude with a discussion of
future prospects for ROM-based quantum computation. We propose a four-qubit
algorithm, using Grover's iterate, for solving a miniature ``real-world''
problem relating to the lengths of paths in a network.Comment: 11 pages, 5 figure
Quantum Algorithm for SAT Problem and Quantum Mutual Entropy
It is von Neumann who opened the window for today's Information epoch. He
defined quantum entropy including Shannon's information more than 20 years
ahead of Shannon, and he introduced a concept what computation means
mathematically. In this paper I will report two works that we have recently
done, one of which is on quantum algorithum in generalized sense solving the
SAT problem (one of NP complete problems) and another is on quantum mutual
entropy properly describing quantum communication processes.Comment: 19 pages, Proceedings of the von Neumann Centennial Conference:
Linear Operators and Foundations of Quantum Mechanics, Budapest, Hungary,
15-20 October, 200
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